Eigenvalues of Maximal Abelian Covers

TL;DR

This paper characterizes the eigenvalues of the maximal abelian cover of finite multi-graphs, solving a longstanding problem and confirming a conjecture about the absence of eigenvalues in regular cases.

Contribution

It provides a complete combinatorial characterization of eigenvalues for maximal abelian covers and proves that regular multi-graphs' covers have no eigenvalues, confirming a conjecture.

Findings

Eigenvalues are characterized by the base graph's combinatorics.

Maximal abelian covers of regular multi-graphs have no eigenvalues.

The criterion relates to existing universal cover eigenvalue criteria.

Abstract

We fully characterize the eigenvalues (flat bands) of the maximal abelian cover of a finite multi-graph in terms of the combinatorics of the base graph. This solves a problem of Higuchi and Nomura (2009, Problem 6.11). We use our new criterion to prove that the maximal abelian cover of any regular multi-graph has no eigenvalues, thereby proving a conjecture of (ibid., Conjecture 6.12). In an appendix, we relate our criterion for eigenvalues of the maximal abelian cover to an existing criterion for eigenvalues of the universal cover.